Algebra Know-It-ALL

366 Quadratic Equations with Real Roots

Are you confused?

Suppose you see an equation in one variable. You think it’s a quadratic, but you aren’t sure. Here’s an

example:

3 x^2 +x− 8 = 4 x^2 + 7 x+ 4

If you can convert this equation to polynomial standard form, you can be certain that it’s a quadratic. If

you can’t convert it, then two things are possible: you didn’t try hard enough, or it isn’t a quadratic. If it

isn’t a quadratic and you want to prove that it isn’t, you must morph the equation into a form that’s obvi-

ously not convertible into polynomial standard form. That can be tricky. The next “challenge” will show

you an example.

As things work out, you can convert the above equation to polynomial standard form. If you subtract

4 x^2 from each side, then subtract 7x from each side, and finally subtract 4 from each side, you get

−x^2 − 6 x− 12 = 0

Here’s a challenge!

Show that the following is the equivalent of a first-degree equation, not a quadratic:

7 x^2 /2+ 7 x− 5 =− 23 x^2 /(−10)+ 6 x^2 /5+ 3 x

Solution

It takes a little intuition to solve this, but nothing beyond the mathematical skill we’ve acquired by now!

Let’s look closely at the right side of this equation. It contains two terms with x^2. These terms are:

− 23 x^2 /(−10)

and

6 x^2 /5

We can add these two terms to get a single term in x^2. If we multiply both the numerator and denominator

of the first of the above expressions by −1, and if we multiply both the numerator and denominator of the

second expression by 2, we get

23 x^2 /10

and

12 x^2 /10

We now have a common denominator, so we can find the sum

23 x^2 /10+ 12 x^2 /10= 35 x^2 /10